Computing Vergne polarizing subalgebras

Abstract

According to the orbit method, the construction of a unitary irreducible representation of a nilpotent Lie group requires a precise computation of some polarizing subalgebra subordinated to a linear functional in the linear dual of the corresponding Lie algebra. This important step is generally challenging from a computational viewpoint. In this paper, we provide an algorithmic approach to the construction of the well-known Vergne polarizing subalgebras. The algorithms presented in this paper are specifically designed so that they can be implemented in computer algebra systems. We also show there are instances where Vergne’s construction could be refined for the sake of efficiency. Finally, we adapt our refined procedure to free nilpotent finite-dimensional Lie algebras of step-two to obtain simple and precise descriptions of Vergne polarizing algebras corresponding to all linear functionals in a dense open subset of the linear dual of the corresponding Lie algebra. Also, a program written for Mathematica is presented at the end of the paper.

Keywords:

• polarizing
• subalgebra
• nilpotent

AMS Subject Classifications:

• 22E25
• 22E27

Acknowledgements

We thank Michel Duflo for bringing to our attention that Niels Pedersen has already written programs in REDUCE to compute polarizing subalgebras for all nilpotent Lie algebras of dimensions less than seven, and we also thank him for supplying Ref. [1].